reserve e for set;
reserve C,C1,C2,C3 for AltCatStr;
reserve C for non empty AltCatStr,
  o for Object of C;
reserve C for non empty transitive AltCatStr;

theorem Th31:
  for C being non empty AltCatStr, D being non empty SubCatStr of
C, o1,o2 being Object of C, p1,p2 being Object of D st o1 = p1 & o2 = p2 holds
  <^p1,p2^> c= <^o1,o2^>
proof
  let C be non empty AltCatStr, D be non empty SubCatStr of C, o1,o2 be Object
  of C, p1,p2 be Object of D such that
A1: o1 = p1 & o2 = p2;
  [p1,p2] in [:the carrier of D, the carrier of D:] & the Arrows of D cc=
  the Arrows of C by Def11;
  hence thesis by A1;
end;
